Mayan Numbers.
The three most common numerical systems in use in the world are all derived from man’s anatomy. The quinary system is based on counting the fingers of one hand, the decimal system on counting those of both hands and the vigesimal system, which prevailed in Central America, is based on counting all the fingers and all the toes. The vigesimal system is seen in imperfect form in our count of scores, where seventy years are three score and ten.
The Mayan name for one was hun: they had simple names to 9 and composite ones from 10 to 19, much as in English, and twenty was hun kal, one score. The ascending values in the vigesimal scale were as follows:—
| Mayan Numbers | Arabic Equivalents | ||
|---|---|---|---|
| hun | 1 | ||
| 20 hun | = 1 | kal | 20 |
| 20 kal | = 1 | bak | 400 |
| 20 bak | = 1 | pic | 8,000 |
| 20 pic | = 1 | cabal | 160,000 |
| 20 cabal | = 1 | kinchil | 3,200,000 |
| 20 kinchil | = 1 | alau | 64,000,000 |
| 20 alau | = 1 | hablat | 1,280,000,000 |
They invented signs for zero and discovered the principle of “local value” in the writing down of numbers centuries before these ideas (which are fundamental to higher mathematics) were known in the Old World. The notation of numbers had its simpler and more complicated phase. In the simpler phase 1 was represented by a dot, 2 by two dots, 5 by a bar, 6 by a bar and dot, 15 by three bars, etc. The commonest sign for zero was a shell while a picture of the moon stood for twenty. In the more elaborate notation a series of twenty faces of gods represented the numerals from 0 to 19.
Fig. 42. Bar and Dot Numerals of the Mayas.
The straight vigesimal system was doubtless used by the Mayas in ordinary counting, but in counting time a very important change was introduced in the third position. Also the names were modified: hun was called kin which means sun or day. In the second position kal was called uinal which means month and 18 of these were taken to form a tun, stone, which was the third unit. The tun then had a value of 18 × 20 = 360 days, making a conventional year about five and a quarter days less than a true year. Twenty tuns made a kaltun or katun and above this period the numeral system proceeded as before and in the ascending values the names already given were merely combined with tun, if Gates is right in his clever suggestion. For years it has been customary to speak of the fifth period as cycle for want of a native term: this will now be called baktun. One hablatun, the highest period with a name, has the astonishing value of 460,800,000,000 days. However, the highest numbers fall considerably short of this potential limit.
In our decimal system the number 347,981, for instance, is really:—
| 3 × | 100000 |
| 4 × | 10000 |
| 7 × | 1000 |
| 9 × | 100 |
| 8 × | 10 |
| 1 × | 1 |
When written out in a horizontal line each “position” has a value ten times that of the “position” to the right of it. It is understood that a digit which stands in a “position” is to be multiplied by 1, 10, 100, 1000, etc., as the case may be. The Mayas, using the principle of position, ordinarily write their bar and dot numerals in columns. But we can partially transcribe a Mayan number in imitation of our own system by putting dots or dashes between the positions or periods. The number in five positions given below is transcribed as 9.12.16.7.8.
| 9 × | 144000 | 1,296,000 |
| 12 × | 7200 | 86,400 |
| 16 × | 360 | 5,760 |
| 7 × | 20 | 140 |
| 8 × | 1 | 8 |
| 1,388,308 |
We read this date: 9 baktuns, 12 katuns, 16 tuns, 7 uinals, and 8 kins. It is convenient to remember that a tun is a little less than a year, a katun a little less than 20 years and a baktun a little less than 400 years. But the count is really of days, not years.
Fig. 43. Face Numerals found in Mayan Inscriptions. In most cases these are the faces of gods. Reading from left to right: the values are 1, 3, 4, 5, 6, 9, 10.
Fig. 44. The Normal Forms of the Period Glyphs. Reading from left to right: baktun, katun, tun, uinal, kin.
Although the numerical values are expressed by position alone in some cases, in other cases use is made of Period Glyphs to make assurance doubly sure. These Period Glyphs represent the basic value of the positions which are to be multiplied by the accompanying numerals. For examples, see Figs. [44] and [45].
Fig. 45. Face Forms of Period Glyphs. From left to right: introducing glyph, baktun, katun, tun, uinal, kin.
[Plate XXIII.]
Typical Mayan Inscription.
Introducing Glyph Initial Series 1. 9 baktuns (cycles). 2. 14 katuns 3. 13 tuns (written 12 by error) 4. 4 uinals 5. 17 kins 6. 12 Caban (day) Supplementary Series 7. glyph F 8. (a) glyph D, (b) glyph C 9. (a) glyph X, (b) glyph B 10. (a) glyph A (30 day lunar month) 10. (b) 5 Kayab (month) Explanatory Series 11, 12, 13 and 14a, possibly explain the dates Secondary Series 14b, 3 kins, 13 uinals 15a, 6 tuns (to be added) Period Ending Date 16. 4 Ahau 13 Yax (9.15.0.0.0)